Centralizers of coprime automorphisms of finite groups

Let $A$ be an elementary abelian group of order $p^k$ with $k\ge 3$ acting on a finite $p^{\mathrm{\prime }}$-group $G$. The following results are proved. If ${\gamma }_{k-2}(C_G(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$,then$\gamma_{k-2}(G)$isnilpotentandhas${c,k,p}$-bounded nilpotency class. If, for some integer $d$ such that $2^d+2\le k$, the $d$th derived group of $C_G(a)$ is nilpotent of class at most $c$ for any a\in A^{#}, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\le 4$.